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Unformatted text preview: STAT 8260 Exam 2 — Tuesday, April 10
SHOW ALL WORK Name: (4) 24‘ e. 1. Suppose we take a random sample of n r» 6 adult fatherson pairs and we measure
the height of each man. Suppose that 3 of the sons in this sample were ﬁrstborn
children in their family and the other 3 happened to be secondborn. Let (yij, as”)
by the (son’s height, father height) pair of measurements corresponding to the jth
son of the ith birth order, 2' = 1, 2, j = 1,2, 3. Consider the following ﬁve models
for such data. yij 0H + eij )
) yij =a+ﬁ$ij+eij yij = 0.1 + £3,113.55, + 81;];
) yij:#+051r(i:2)+ﬁ$ij+eij
) yij=H+a1(i=2)+ﬁ$,jI(i:l)+7x,jI(i=2)+eij where I denotes an indicator variable taking the value 1 when the condition inside the parentheses is true and the value 0 otherwise. All models have the same
assumptions on the eijS: 811,...623 trad N(0, 02). a. (8 pts.) Which, if any, of these models are equivalent. ((1?) (V) W1 éfVIl’ﬁ/‘ﬁi )g 1%)0 are a” yw gay/J M54 aim % m e/ marézanﬂ 4// 5 m/d, M) (Luz WAIOL one! Awe >g/amc CD/zmm Vacs 24/ f MAJ!
7%: 7; J' MV‘Z‘ gar/u %’" MoJel Veil{(2.3 2 [WI , 0?, a.) 04, 74; >46 ijmyf,rt5/oeq4m9. ﬂock, (if) Eyeaids
a 5245/! fej/gS’ﬁBn /4¢ /Lutm~o4 lavécuf/i/EJIA/ﬁ/i) ﬁ‘ q//fvéj‘a713,
/7ao/el (at) a/Aur j» reg/05% Ad, 0653% 74 5m? /J “Add/9%
75/ 5M)” 2  #7de (5") 5 ear/u 2 alt/Kai «aka/797k ‘49
/M+o( A,“ ﬂaw/U f9“) iuf‘c‘reagnb/ “3/6 le‘tdn. fé/L /“ J./ (v) Aka not/ﬂ (:22) 57°04; c/erjkc/Ihﬁyé/k b1)
fIAJ/Jl ﬁt J / V4 %
Whig 7/, ,15 2 gnu/33 4.4) Jar/{457’ Singﬂﬂ “'3 Y)
) 1 2. éfw S b. (9 pts.) Write down model (iii) in vector/ matrix notatiOn. ‘3“ I o Xu 0 “(I 2“
ll. “3‘1 o X” 0 dz 1L CI? “3:? : i o X13 0 f: 62‘ 22»! O l 0 X“ 824. L o l 0 K11. ﬂ;
‘3‘" 0 t (9 XL} 613
3’55 c. (9 pts.) Under the maintained hypothesis that model (v) holds, express
the hypothesis that H0 : {model (ii) holds}, in the form of the general linear hypothesis on the parameters of model ﬂ
0 l 0 O 
C :1 Il, pale/€— C 2 VB “ °’
2f d. (9 pts.) Based upon the results in the table below, compute an appropriate
test statistic (give me its numeric value for these data) and give its reference
distribution for testing the hypothesis in part Model MSE R2
(ii) 9.9268 .0432
(v) 3.2262 .8445 [federal W? 74 Ann/v7: F)
a 6; ‘ A?” : (LIE7W} (\ T:[A, nk") ; (.Ql‘lﬁﬂBQ/L F/ZIZ) UAJQ/ <1, .K‘Hﬂ/g—z— :) : 5.7;, 2. Consider the regression model 911 1 611
9'12 1 612
2 +
@121 ﬁ 1 621
922 1 622
= [3.14 + e
where (811) N N (0,0? (I p)) independent of (621) N N(0,0'§I) where
612 p 1 622 03 = 20? and p = .5. a. (13 pts.) Derive a simple, nonmatrix formula for BBLUE, the best linear unbiased estimator of ﬂ in this model. I , g 0 o
6~N{o,6,"’l/) aim V: 29 cf
/\ a 40 0 ‘9 2 r
Engf—(S % agn‘ﬂr ()(TV “14,1 4 ‘f ' " O r 3 s .1 .1) Ami/ab,“
v= a w
.1. 0 “‘4— 3
9 ﬂ; ‘4) (m> a:
ﬁ —l l i (1'. 3 .3 j (J‘TVJJIJ’VJ " % E 1‘: j 1 '72 ¥)/l1)/'7 _
_'_3_
:5 4' N ‘35" b. (13 pts.) Assuming the model is correct, compute the ratio Y \ =fl=‘{ val:(g‘.)
WWWBLUE)a
where 3].. = ﬁzz2:1 23:1 yij is the sample mean.
‘57.. w 245 am any/M /) MK WM“
_.  o "\ — " ' '
Wm): We"ersz =91) JEWEL).
2, .
... _L. I ... 3 .3. \ ) J VA” 7:) LJZJZ)‘J— :31.
._ 1' 7' M,
_(5_\ /z _ Sr 03 Tle‘l 30—?3/¥
A __ TV :
\(oJ(/3I3Lwr) ’ 0—: (X X I J a
—— J 3 W
3 Varbﬁ 3 I H: __ _,___._
:1 4 2 a 3— "" Q
may» a; g, 7:1
4075'“ > I 3. Consider the classical linear model
y = Xﬁ + e, e N Nn(0,a21n) where X is n x (Is + 1) with rank(X) = k + 1 < n. Suppose this model is ﬁt to data y yielding the ordinary least squares regression parameter estimate [3 and.
MSE, 32. We wish to form a prediction interval for an additional observation yo
conforming to this model. That is, yo = xiii? + 60 Where am is the a vector of
explanatory variables corresponding to yo (x0 can be considered as an additional row of X) and 60 w N(0, 02) independent of e.
(13 pts.) For 330 x x313, show that 310330 8V 1 +XE(XTX)_1X0 N150” W k — 1). j \
1.
4 1. l . film/UK;— if”
5W gIvN/ﬂ, Mm Ma‘s ’ ’ 4. Consider the classical linear model yzxﬁ‘l'e: ewNn(0702In) Where X is n x p with rank(X) = p < n. Deﬁne (33 = SSE/(n—E), an estimator of
02. Recall the mean square error of an estimator T of a parameter 9 is E{(T ~6)2} . . 2
but can also be represented as varlance + blas . a. (13 pts.) Find the mean square error of 6'3, for an arbitrary value of the constantﬂ.
,_.'7
Var 332;): (All); WM?“ ’Cm‘é’l
‘ ~—’ Erma: ewe/(KW); HM
_. @rl)2 =0
1 Z 1:5 (:5
— (rtif “we” 9“” 2
‘— ﬁ (1 2. ’_
,BFM(E¢2): ,Bms(n—o §§c):[(m_£$3 (5'
re n") (IA‘ 2 z A“
=S“ :_ 11 £152 (5 : n“!
M T
 NZ __ n— L! (1—. Z EgzZOjIO” FEED b. (13 pts.) Find the value of E that yields the estimator Er? with the smallest _ ‘i
mean square error f ~2.) ’ W + q r 2090143) (“015 4'
4 52+
H0“ (11’?) 9470179} _. : O (“gigs
M? W} (WMMWJL 5’ 2m»
3 l +(nf>(nnufe" :0
1:) :2 + (m? M3 :0 :51 ZPJ
SSE 1r 7%5 MMIMUM AME a574M£f
:3 aniv'a a? of: ...
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